Study on Transformation Group Theory and Equivariant K-theory

变换群理论和等变K理论研究

• 批准号: 12640075
• 负责人: KOMIYA Katsuhiro
• 金额: $ 0.9万
• 依托单位: Yamaguchi University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640075/
• 关键词: Borsuk-Ulam theorem; linear representation; equivariant map; equivariant K-theory; cutting and pasting; SK-equivalence; SK-group; Euler characteristic; cutting; pasting; Z_2多様体; double; 境界をもつ多様体; Borelコホモロジー理論; 写像度; 表現環

[1] The classical Borsuk-Ulam theorem is only concerned with equivariant maps between spheres with the antipodal action of the cyclic group of order 2. In our study we generalize this to equivariant maps between unit spheres of unitary representations of a compact abelian group. We obtain a necessary and sufficient condition for the existence of such equivariant maps in terms of the Euler classes of unit spheres, and also obtain the result concerning with the degrees of equivariant maps. The method depends heavily on the equivariant K-theory.[II] Cuttings and pastings of manifolds lead to the so-called SK-groups of manifolds. Various kinds of such groups are found in the literature. In our study we investigate the relation among them, and show that the following three SK-groups are essentially isomorphic to each other : (i) the SK-group of pairs of manifolds, (ii) the SK-group of manifolds with boundary, and (iii) the SK-group of manifolds with involution.

[1]经典的Borsuk-Ulam定理只涉及具有2阶循环群的对极作用的球面之间的等变映射。在我们的研究中，我们将其推广到紧交换群的酉表示的单位球面之间的等变映射。利用单位球面的欧拉类得到了这种等变映射存在的充要条件，并得到了与等变映射的次数有关的结果。这种方法在很大程度上依赖于等变K-理论。流形的切割和粘贴导致了流形的所谓SK-群。在文献中可以找到各种各样的这样的群体。在我们的研究中，我们研究了它们之间的关系，并证明了以下三个SK-群本质上是彼此同构的：(I)流形对的SK-群，(Ii)有边界流形的SK-群，(Iii)有对合流形的SK-群。

项目成果

Katsuhiro Komiya: "Introduction to Topology"Shokabo. 152 (2001)

小宫胜宏：《拓扑学概论》Shokabo。

Komiya Katsuhiro: "Cutting, pasting and the doubles of manifolds with boundary"Kyushu Journal of Mathematics. 55. 321-328 (2001)

小宫胜宏：“剪切、粘贴和有边界的流形的双打”九州数学杂志。

Katsuhiro Komiya: "Cutting-pasting and doubles of manifolds with boundary"Kyushu Journal of Mathematics. 55-1(or-2)(近刊). (2001)

Katsuhiro Komiya：“剪切粘贴和带边界的双倍”《九州数学杂志》55-1（或-2）（即将出版）。

Katsuhiro Komiya: "Equivariant K-theoretic Euler classes and maps of representation spheres"Osaka Journal of Mathematics. 38. 239-249 (2001)

Katsuhiro Komiya：“等变K-理论欧拉类和表示球面图”大阪数学杂志。

Katsuhiro Komiya: "Cutting, pasting and the doubles of manifolds with boundary"Kyushu Journal of Mathematics. 55. 321-328 (2001)

小宫胜宏：“剪切、粘贴和带边界的流形的双打”九州数学杂志。